I have no idea what you mean by “installing the Arch way” but here is the package list you get when you install budgie-desktop on Arch.
pacman -S budgie-desktop ±[●][zfs-wip]
resolving dependencies...
looking for conflicting packages...
Packages (29) alsa-plugins-1:1.2.5-2 bubblewrap-0.5.0-1 enchant-2.3.1-1 gcr-3.40.0-1 geoclue-2.5.7-4 geocode-glib-3.26.2-1 gnome-bluetooth-3.34.5-2 gnome-desktop-1:40.4-1 gnome-menus-3.36.0-1 gnome-session-40.1.1-1 gnome-settings-daemon-40.0.1-2 gnome-themes-extra-3.28-2
gobject-introspection-runtime-1.70.0-2 harfbuzz-icu-3.1.0-1 libgweather-40.0-1 libmanette-0.2.6-2 libpeas-1.30.0-1 libwnck3-40.0-1 libwpe-1.12.0-1 mutter-40.5-1 pulseaudio-alsa-1:1.2.5-2 pulseaudio-bluetooth-15.0-1 startup-notification-0.12-7
webkit2gtk-2.34.1-2 wpebackend-fdo-1.12.0-1 xdg-dbus-proxy-0.1.2-3 xorg-xwayland-21.1.2.901-1 zenity-3.41.0-1 budgie-desktop-10.5.3-1
As you can see, gnome-session is one of the dependencies so you will get a working gnome session if you install budgie on an Arch-based distro.
Further, most budgie users will probably also want gnome-control-center. This will bring in more of gnome.
pacman -S budgie-desktop gnome-control-center ±[●][zfs-wip]
resolving dependencies...
looking for conflicting packages...
Packages (117) aalib-1.4rc5-14 alsa-plugins-1:1.2.5-2 bubblewrap-0.5.0-1 cheese-3.38.0-1 chromaprint-1.5.0-3 clutter-1.26.4-1 clutter-gst-3.0.27-3 clutter-gtk-1.8.4-2 cogl-1.22.8-1 colord-1.4.5-4 colord-gtk-0.2.0-2 cracklib-2.9.7-2 cups-pk-helper-0.2.6-4 enchant-2.3.1-1
faac-1.30-2 faad2-2.10.0-1 fftw-3.3.10-1 fluidsynth-2.2.3-1 frei0r-plugins-1.7.0-1 gavl-1.4.0-5 gcr-3.40.0-1 geoclue-2.5.7-4 geocode-glib-3.26.2-1 gnome-bluetooth-3.34.5-2 gnome-color-manager-3.36.0-1 gnome-desktop-1:40.4-1 gnome-menus-3.36.0-1
gnome-online-accounts-3.40.1-1 gnome-session-40.1.1-1 gnome-settings-daemon-40.0.1-2 gnome-themes-extra-3.28-2 gnome-video-effects-0.5.0+4+g9554041-2 gobject-introspection-runtime-1.70.0-2 grilo-0.3.14-1 gsound-1.0.3-1 gssdp-1.4.0.1-1 gst-plugins-bad-1.18.5-3
gst-plugins-bad-libs-1.18.5-3 gst-plugins-good-1.18.5-2 gupnp-1:1.4.0-1 gupnp-igd-1.2.0-2 harfbuzz-icu-3.1.0-1 hwloc-2.5.0-1 imagemagick-7.1.0.13-1 imath-3.1.3-1 imlib2-1.7.4-1 libavtp-0.1.0-2 libbs2b-3.1.0-7 libcaca-0.99.beta20-1 libdc1394-2.2.6-2
libdca-0.0.7-1 libde265-1.0.8-1 libdv-1.0.0-9 libdvdnav-6.1.1-1 libdvdread-6.1.2-1 libgme-0.6.3-1 libgnomekbd-3.26.1+2+g8d02ebd-2 libgtop-2.40.0+2+g31db82ef-2 libgusb-0.3.8-1 libgweather-40.0-1 libhandy-1.4.0-1 libid3tag-0.15.1b-11 libinstpatch-1.1.6-1
libkate-0.4.1-7 liblqr-0.4.2-3 liblrdf-0.6.1-4 libmanette-0.2.6-2 libmicrodns-0.2.0-1 libmms-0.6.4-3 libmpcdec-1:0.1+r475-3 libnice-0.1.18-2 libnma-1.8.32-1 liboauth-1.0.3+14+g186fb14-2 libofa-0.9.3-9 libpeas-1.30.0-1 libpwquality-1.4.4-3 libraqm-0.7.2-2
libshout-1:2.4.5-1 libsrtp-1:2.4.2-1 libwnck3-40.0-1 libwpe-1.12.0-1 libxklavier-5.4-3 lilv-0.24.12-1 mjpegtools-2.2.1-1 mpg123-1.29.2-1 mutter-40.5-1 neon-0.32.1-1 nm-connection-editor-1.24.0-1 openexr-3.1.3-1 openmpi-4.1.1-3 portaudio-1:19.7.0-1
pulseaudio-alsa-1:1.2.5-2 pulseaudio-bluetooth-15.0-1 raptor-2.0.15-16 rtmpdump-1:2.4.r96.fa8646d-6 serd-0.30.10-1 shared-color-targets-0.1.7-3 sord-0.16.8-1 soundtouch-2.3.0-1 spandsp-0.0.6-3 sratom-0.6.8-1 startup-notification-0.12-7 svt-hevc-1.5.1-1
totem-pl-parser-3.26.6-1 twolame-0.4.0-2 wavpack-5.4.0-1 webkit2gtk-2.34.1-2 wildmidi-0.4.4-1 wpebackend-fdo-1.12.0-1 xdg-dbus-proxy-0.1.2-3 xorg-xwayland-21.1.2.901-1 zbar-0.23.1-7 zenity-3.41.0-1 zvbi-0.2.35-4 zxing-cpp-1.2.0-1 budgie-desktop-10.5.3-1
gnome-control-center-40.0-1
Neither option brings in every application that could be included with gnome or is in the gnome group but nobody claimed that.